61,365 research outputs found

### Free energy of the three-state $\tau_2(t_q)$ model as a product of elliptic functions

{We show that the free energy of the three-state
$\tau_2(t_q)$ model can be expressed as products of Jacobi elliptic
functions, the arguments being those of an hyperelliptic parametrization of the
associated chiral Potts model. This is the first application of such a
parametrization to the $N$-state chiral
Potts free energy problem for $N > 2$.Comment: 20 pages, 3 figure

### Some exact results for the three-layer Zamolodchikov model

In this paper we continue the study of the three-layer Zamolodchikov model
started in our previous works. We analyse numerically the solutions to the
Bethe ansatz equations. We consider two regimes I and II which differ by the
signs of the spherical sides (a1,a2,a3)->(-a1,-a2,-a3). We accept the two-line
hypothesis for the regime I and the one-line hypothesis for the regime II. In
the thermodynamic limit we derive integral equations for distribution densities
and solve them exactly. We calculate the partition function for the three-layer
Zamolodchikov model and check a compatibility of this result with the
functional relations. We also do some numerical checkings of our results.Comment: LaTeX, 27 pages, 9 figure

### Algebraic reduction of the Ising model

We consider the Ising model on a cylindrical lattice of L columns, with
fixed-spin boundary conditions on the top and bottom rows. The spontaneous
magnetization can be written in terms of partition functions on this lattice.
We show how we can use the Clifford algebra of Kaufman to write these partition
functions in terms of L by L determinants, and then further reduce them to m by
m determinants, where m is approximately L/2. In this form the results can be
compared with those of the Ising case of the superintegrable chiral Potts
model. They point to a way of calculating the spontaneous magnetization of that
more general model algebraically.Comment: 25 pages, one figure, last reference completed. Various typos fixed.
Changes on 12 July 2008: Fig 1, 0 to +1; before (2.1), if to is; after (4.6),
from to form; before (4.46), first three to middle two; before (4.46), last
to others; Conclusions, 2nd para, insert how ; renewcommand \i to be \rm

### The challenge of the chiral Potts model

The chiral Potts model continues to pose particular challenges in statistical
mechanics: it is ``exactly solvable'' in the sense that it satisfies the
Yang-Baxter relation, but actually obtaining the solution is not easy. Its free
energy was calculated in 1988 and the order parameter was conjectured in full
generality a year later.
However, a derivation of that conjecture had to wait until 2005. Here we
discuss that derivation.Comment: 22 pages, 3 figures, 29 reference

### Derivation of the order parameter of the chiral Potts model

We derive the order parameter of the chiral Potts model, using the method of
Jimbo et al. The result agrees with previous conjectures.Comment: Version 2 submitted 21 Feb 2005. It has 7 pages, 2 figures. The
introduction has been expanded and a significant typographical error in eqn
23 has been correcte

### The Large N Limits of the Chiral Potts Model

In this paper we study the large-N limits of the integrable N-state chiral
Potts model. Three chiral solutions of the star-triangle equations are derived,
with states taken from all integers, or from a finite or infinite real
interval. These solutions are expected to be chiral-field lattice deformations
of parafermionic conformal field theories. A new two-sided hypergeometric
identity is derived as a corollary.Comment: 41 pages, 3 figures, LaTeX 2E file, using elsart.cls and psbox.tex
(version 1.31 provided), [email protected]

### The order parameter of the chiral Potts model

An outstanding problem in statistical mechanics is the order parameter of the
chiral Potts model. An elegant conjecture for this was made in 1983. It has
since been successfully tested against series expansions, but as far as the
author is aware there is as yet no proof of the conjecture. Here we show that
if one makes a certain analyticity assumption similar to that used to derive
the free energy, then one can indeed verify the conjecture. The method is based
on the ``broken rapidity line'' approach pioneered by Jimbo, Miwa and
Nakayashiki.Comment: 29 pages, 7 figures. Citations made more explicit and some typos
correcte

### Corner transfer matrices in statistical mechanics

Corner transfer matrices are a useful tool in the statistical mechanics of
simple two-dimensinal models. They can be very effective way of obtaining
series expansions of unsolved models, and of calculating the order parameters
of solved ones. Here we review these features and discuss the reason why the
method fails to give the order parameter of the chiral Potts model.Comment: 18 pages, 4 figures, for Proceedings of Conference on Symmetries and
Integrability of Difference Equations. (SIDE VII), Melbourne, July 200

### Planar lattice gases with nearest-neighbour exclusion

We discuss the hard-hexagon and hard-square problems, as well as the
corresponding problem on the honeycomb lattice. The case when the activity is
unity is of interest to combinatorialists, being the problem of counting binary
matrices with no two adjacent 1's. For this case we use the powerful corner
transfer matrix method to numerically evaluate the partition function per site,
density and some near-neighbour correlations to high accuracy. In particular
for the square lattice we obtain the partition function per site to 43 decimal
places.Comment: 16 pages, 2 built-in Latex figures, 4 table

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